The implications of causality on the behavior of the constitutive parameters in the time domain can be easily identified. Writing(2.29)and(2. 31)after setting u=t-t and then u=t we have D(r, t)=EoE(r, t)+Eo/Xe(r, tE(r, t-t)dt J(r,t)=o(r,!E(r, t-I)dt We see that there is no contribution from values of xe(r, t) or o(r, t) for times t <0. So we can write Dr,)=∈0E(r,1)+60/x(r,t)E(r,t-)d' J(r, t)= o(r, t'E(r, t -)dr' th the additional assumption Xe(r, t)=0, (r,t)=0.t<0. (4.30) By(4.30) we can write the frequency-domain complex permittivity(4.26)as a(r, t')e-jonr'dr'+Eo/Xe(r,t' In order to derive the Kronig-Kramers relations we must understand the behavior of E(r, o)-Eo in the complex a-plane. Writing o= Or+ joi, we need to establish the following two properties Property 1: The function E(r, a)-Eo is analytic in the lower half-plane(o; 0) ept at o=0 We can establish the analyticity of o(r, o) by integrating over any closed contour in the lower half-plane. We have (r, o)do σ(r,t Note that an exchange in the order of integration in the above expression is only valid for o in the lower half-plane where limr'-oo e /or=0. Since the function f(o)=e- Jor is analytic in the lower half-plane, its closed contour integral is zero by the Cauchy-Goursat theorem. Thus, by(4.32)we have Then, since a may be assumed to be continuous medium, and since its closed path integral is zero for ible paths T, it is by Morera's theorem [110 analytic in the lower half-plane. By reasoning xe (r, o) is analytic in the lower half-plane. Since the function 1/o has a simple pole at o=0, the composite function E(r, a)-Eo given by(4.31)is analytic in the lower half-plane excluding o=0 here it has a simple pole 2001 by CRC Press LLCThe implications of causality on the behavior of the constitutive parameters in the time domain can be easily identified. Writing (2.29) and (2.31) after setting u = t − t and then u = t , we have D(r, t) = 0E(r, t) + 0 ∞ 0 χe(r, t  )E(r, t − t  ) dt , J(r, t) = ∞ 0 σ(r, t  )E(r, t − t  ) dt . We see that there is no contribution from values of χe(r, t) or σ(r, t) for times t < 0. So we can write D(r, t) = 0E(r, t) + 0 ∞ −∞ χe(r, t  )E(r, t − t  ) dt , J(r, t) = ∞ −∞ σ(r, t  )E(r, t − t  ) dt , with the additional assumption χe(r, t) = 0, t < 0, σ(r, t) = 0, t < 0. (4.30) By (4.30) we can write the frequency-domain complex permittivity (4.26) as ˜ c (r,ω) − 0 = 1 jω ∞ 0 σ(r, t  )e− jωt dt + 0 ∞ 0 χe(r, t  )e− jωt dt . (4.31) In order to derive the Kronig–Kramers relations we must understand the behavior of ˜ c(r,ω) − 0 in the complex ω-plane. Writing ω = ωr + jωi , we need to establish the following two properties. Property 1: The function ˜ c(r,ω) − 0 is analytic in the lower half-plane (ωi < 0) except at ω = 0 where it has a simple pole. We can establish the analyticity of σ(˜ r,ω) by integrating over any closed contour in the lower half-plane. We have   σ(˜ r,ω) dω =    ∞ 0 σ(r, t  )e− jωt dt dω = ∞ 0 σ(r, t  )   e− jωt dω dt . (4.32) Note that an exchange in the order of integration in the above expression is only valid for ω in the lower half-plane where limt →∞ e− jωt = 0. Since the function f (ω) = e− jωt is analytic in the lower half-plane, its closed contour integral is zero by the Cauchy–Goursat theorem. Thus, by (4.32) we have   σ(˜ r,ω) dω = 0. Then, since σ˜ may be assumed to be continuous in the lower half-plane for a physical medium, and since its closed path integral is zero for all possible paths , it is by Morera’s theorem [110] analytic in the lower half-plane. By similar reasoning χe(r,ω) is analytic in the lower half-plane. Since the function 1/ω has a simple pole at ω = 0, the composite function ˜ c(r,ω) − 0 given by (4.31) is analytic in the lower half-plane excluding ω = 0 where it has a simple pole.